Add Momentum SGD optimizer implementation

- Implements SGD with momentum using pure NumPy
- Includes comprehensive docstrings and type hints
- Adds doctests for validation
- Provides usage example demonstrating convergence
- Follows PEP8 coding standards

Author: Adhithya-Laxman

neural_network/optimizers/__init__.py

@@ -0,0 +1,88 @@
+"""
+Momentum SGD Optimizer
+
+Implements SGD with momentum for neural network training using NumPy.
+Momentum helps accelerate gradients in the relevant direction and dampens oscillations.
+
+Reference: https://en.wikipedia.org/wiki/Stochastic_gradient_descent#Momentum
+Author: Adhithya Laxman Ravi Shankar Geetha
+Github: https://github.com/Adhithya-Laxman
+Date: 2025.10.22
+"""
+
+import numpy as np
+
+
+class MomentumSGD:
+    """
+    SGD with momentum optimizer.
+
+    Updates parameters using momentum:
+        velocity = momentum * velocity - learning_rate * gradient
+        param = param + velocity
+    """
+
+    def __init__(self, learning_rate: float = 0.01, momentum: float = 0.9) -> None:
+        """
+        Initialize Momentum SGD optimizer.
+
+        Args:
+            learning_rate (float): Learning rate for weight updates.
+            momentum (float): Momentum factor.
+
+        >>> optimizer = MomentumSGD(learning_rate=0.01, momentum=0.9)
+        >>> optimizer.momentum
+        0.9
+        """
+        self.learning_rate = learning_rate
+        self.momentum = momentum
+        self.velocity: dict[int, np.ndarray] = {}
+
+    def update(
+        self, param_id: int, params: np.ndarray, gradients: np.ndarray
+    ) -> np.ndarray:
+        """
+        Update parameters using momentum.
+
+        Args:
+            param_id (int): Unique identifier for parameter group.
+            params (np.ndarray): Current parameters.
+            gradients (np.ndarray): Gradients of parameters.
+
+        Returns:
+            np.ndarray: Updated parameters.
+
+        >>> optimizer = MomentumSGD(learning_rate=0.1, momentum=0.9)
+        >>> params = np.array([1.0, 2.0])
+        >>> grads = np.array([0.1, 0.2])
+        >>> updated = optimizer.update(0, params, grads)
+        >>> updated.shape
+        (2,)
+        """
+        if param_id not in self.velocity:
+            self.velocity[param_id] = np.zeros_like(params)
+
+        self.velocity[param_id] = (
+            self.momentum * self.velocity[param_id] - self.learning_rate * gradients
+        )
+        return params + self.velocity[param_id]
+
+
+# Usage example
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    print("Momentum SGD Example: Minimizing f(x) = x^2")
+
+    optimizer = MomentumSGD(learning_rate=0.1, momentum=0.9)
+    x = np.array([5.0])
+
+    for step in range(20):
+        gradient = 2 * x
+        x = optimizer.update(0, x, gradient)
+        if step % 5 == 0:
+            print(f"Step {step}: x = {x[0]:.4f}, f(x) = {x[0] ** 2:.4f}")
+
+    print(f"Final: x = {x[0]:.4f}, f(x) = {x[0] ** 2:.4f}")